We have no personal knowledge but see visitors riding mopeds and motorcycles so know there must be businesses that supply them.
If you follow this search link: SEARCH LINK you will see several companies listed with phone numbers -- probably the best thing to do is call and compare what they have and their prices.
One that comes up is Kiser Motorcycle at 74-5622 Alapa Street in Kailua-Kona. Their phone number is 808-329-9100 and their email is kiserhonda@hawaii.rr.com.
Another that comes up is West Coast Scooter at 74-5616 Alapa Street in Kailua-Kona. Their phone number is 808-329-1188.
Another that comes up is Kona Harley Davidson at 74-5615 Luhia St # E in Kailua-Kona. Their phone number is 808-326-9887.
Recently while driving, happened to notice that there was a sign advertising “motorcycles for rent” at Affordable Rent-A-Car on Palani RD. This appears when I just did an internet search to be at 75-5614 Palani Road with phone number 808-329-7766
Also new to me (4/6/09) as you drive on King K highway past Costco there is a banner advertising about motorcycle rentals. We also found a flyer distributed and think this relates to Kona Cost Custom (Kona Coast Custom Website), 73-4776 Kanalani Street, 808-325-2500. Their flyer says they are open 9AM-5PM Monday thru Saturday. Also says some requirements you will need to rent are: be 21, have a valid motorcycle operator’s license, have the skill and ability to operate a heavyweight motorcycle, and have a major credit card.
Last I checked the above, the age is now 25 or over and it is $135 a day.